Answer Happy

Q7) (Theoretical/Practical Question) In this question we develop Yule-Walker estimators in AR(1) and ARMA(1,1) models and study their numerical performance. Recall from lectures that in AR(1) model Xt​=ϕXt−1​+Zt​ the Yule-Walker estimator is ϕ​=γ​X​(0)γ​X​(1)​=ρ​X​(1),σZ2​=γ​X​(0)−ϕ​γX​​(1)=γ​X​(0)−ρ​X​(1)2γ​X​(0). (a) Numerical experiment for AR(1) : * Load into R the file Data-AR.txt. (Just type Data=scan(file.choose()) and then copy and paste). This is data set generated from AR(1) model with ϕ=0.8. * Type var (Data) to obtain γ​X​(0). * Type ACF<-acf (Data). Then type ACF. You will get ρ​X​(h), the estimators of ρX​(h). The second entry will be ρ​X​(1). Via the formula above this is also ϕ​. * Write the final values for ϕ​ and σZ2​. * Compare your estimated ϕ​ with the true ϕ. (b) Consider ARMA (1,1) model Xt​=ϕXt−1​+Zt​+θZt−1​,∣ϕ∣<1, so that the sequence Xt​ is causal. Apply the Yule-Walker procedure to get the estimators for ϕ,θ and σZ2​=Var(Zt​). HINT: You should get ϕ=γX​(1)γX​(2)​,γX​(1)=ϕγX​(0)+θσZ2​,γX​(0)=σZ2​[1+1−ϕ2(θ+ϕ)2​]. (c) Numerical experiment for ARMA(1,1) : * Load into R the file Data-ARMA.txt. (Just type Data=scan(file.choose()) and then copy and paste). This is data set generated from ARMA (1,1) model with ϕ=0.8 and θ=1. * Write the final values for ϕ​,θ^ and σZ2​. * Compare your estimated ϕ​ with the true ϕ. Which estimate is more accurate, for ARMA(1,1) or for AR(1) ?